For the time discretization, an IMplicit EXplicit (IMEX) Additive Runge Kutta method (ARK) [1]} method will be used. These methods are useful for time dependent problems that can be formulated as
\( \mathbf {y}^{\prime } = \mathbf {f}_{S}(\mathbf {y},t) +\mathbf {f}_{NS}(\mathbf {y},t) \) , where the \(S\)  and \(NS\)  subscripts denote the stiff and non-stiff components of the system, to which the implicit and explicit companion methods are applied, respectively.
If \(\mathbf {v}^{n}\approx y(t^n), \)  the generic \( s- \)  stage IMEX-ARK method can be defined as
\(\begin{array}{rcl}\mathbf {v}^{(n,l)} = \mathbf {v}^n &+&\Delta t \sum \limits _{m=1}^{s-1} \bigg ( a_{lm}\mathbf {f}_{NS}(\mathbf {v}^{(n,m)},t+c_m \Delta t) \\[4mm] &+&\tilde{a}_{lm}\mathbf {f}_{S}(\mathbf {v}^{(n,m)},t+c_m\Delta t) \bigg ) + \Delta t \, \tilde{a}_{ll} \, \mathbf {f}_{S}(\mathbf {v}^{(n,l)},t+c_l\Delta t).\end{array}\nonumber \) 
